Archive for the 'QM Method' Category

Pericyclic reactions remain a fruitful area of research despite the seminal publication of the Woodward-Hoffmann rules decades ago. Here are two related papers of pericyclic reactions that violate the Woodward-Hoffmann rules.

First, Solomek, Ravat, Mou, Kertesz, and Jurícek reported on the thermal and photochemical electrocyclization reaction of diphenylcetherene 1a.1 Though they were not able to directly detect the intermediate 2, through careful examination of the photochemical reaction, they were able to infer that the thermal cyclization goes via the formally forbidden conrotatory pathway (see Scheme 1).

Scheme 2.

Kinetic studies estimate the activation barrier is 14.1 kcal mol-1. They performed DFT computations of the parent 1b using a variety of functionals with both restricted and unrestricted wavefunctions. The allowed pathway to 2syn is predicted to be greater than 27 kcal mol-1, while the formally forbidden pathway to 2anti is estimated to have a lower barrier of about 23 kcal mol-1. The two transition states for these different pathways are shown in Figure 1, and the sterics that force a helical structure to 1 help make the forbidden pathway more favorable.

TS(1b→2b-syn)

TS(1b→2b-anti)

Figure 1. (U)B3LYP/6-31G optimized geometries of the transition states taking 1 into 2.

Nonetheless, all of the DFT computations significantly overestimate the activation barrier. The authors make the case that a low-lying singlet excited state results in an early conical intersection that reduces the symmetry from C2 to C1. In this lower symmetry pathway, all of the states can mix, leading to a lower barrier. However, since DFT is intrinsically a single Slater configuration, the mixing of the other states is not accounted for, leading to the overestimated barrier height.

In a follow up study, this group examined the thermal and photo cyclization of 13,14-dimethylcethrene 4.2 The added methyl groups make the centhrene backbone more helical, and this precludes the formal allowed disrotatory process. The methyl groups also prohibit the oxidation that occurs with 1, driven by aromatization, allowing for the isolation of the direct product of the cyclization 5. This anti stereochemistry is confirmed by NMR and x-ray crystallography. The interconversion between 4 and 5 can be controlled by heat and light, making the system an interesting photoswitch.

Also of interest is the singlet-triplet gap of 1 and 4. The DFT computed ΔEST is about 10 kcal mol-1 for 4, larger than the computed value of 6 kcal mol-1 for 1b. The EPR of 1b does show a signal while that of 4 has no signal. To assess the role of the methyl group, they computed the singlet triplet gaps for 1b and 4 at two different geometries: where the distance between the carbons bearing the methyl groups is that in 1b (3.03 Å) and in 4 (3.37 Å). The lengthening of this distance by the methyl substituents is due to increased helical twist in 4 than in 1b. For 1b, the gap increases with twisting, from 7.1 to 8.3 kcal mol-1, while for 4 the gap increases by 1.8 kcal mol-1 with the increased twisting. This change is less than the effect of methyl substitution, which increases the gap by 2.2 kcal mol-1 at the shorter distance and 2.8 kcal mol-1 at the longer distance. Thus, the electronic (orbital) effect of methyl substitution affects the singlet-triplet gap more than the geometric twisting.

Schreiner and Grimme have examined a few compounds (see these previous posts) with long C-C bonds that are found in congested systems where dispersion greatly aids in stabilizing the stretched bond. Their new paper1 continues this theme by examining 1 (again) and 2, using computations, and x-ray crystallography and gas-phase rotational spectroscopy and electron diffraction to establish the long C-C bond.

The distance of the long central bond in 1 is 1.647 Å (x-ray) and 1.630 Å (electron diffraction). Similarly, this distance in 2 is 1.642 Å (x-ray) and 1.632 Å (ED). These experiments discount any role for crystal packing forces in leading to the long bond.

A very nice result from the computations is that most functionals that include some dispersion correction predict the C-C distance in the optimized structures with an error of no more than 0.01 Å. (PW6B95-D3/DEF2-QZVP structures are shown in Figure 1.) Not surprisingly, HF and B3LYP without a dispersion correction predict a bond that is too long.) MP2 predicts a distance that is too short, but SCS-MP2 does a very good job.

Click chemistry has been used in a broad range of applications. The use of metal catalysts has limited its application to biological system, but the development of strain-promoted cycloaddition to cyclooctyne has opened up click chemistry to bioorthogonal labeling.

An interesting variation on this is the use of 1,2-benzoquinone 1 and substituted analogues as the Diels-Alder diene component. Escorihuela and co-workers have reported on the use of this diene with a number of cyclooctyne derivatives, measuring kinetics and also using computations to assess the mechanism.1

Their computations focused on two reactions using cyclooctyne 2 and the cyclopropane-fused analogue 3:

Reaction 1

Reaction 2

They examined these reactions with a variety of density functionals along with some post-HF methods. The transition states of the two reactions are shown in Figure 1. A variety of different density functionals and MP2 are consistent in finding synchronous or nearly synchronous transition states.

Rxn1-TS

Rxn2-TS

Figure 1. B97D/6-311+G(d,p) transition states for Reactions 1 and 2.

In terms of activation energies, all of the DFT methods consistently overestimate the barrier by about 5-10 kcal mol-1, with B97D-D3 doing the best. MP2 drastically underestimates the barriers, though the SOS-MP2 or SCS-MP2 improve the estimate. Both CCSD(T) and MR-AQCC provide estimates of about 8.5 kcal mol-1, still 3-4 kcal mol-1 too high. The agreement between CCSD(T), a single reference method, and MR-AQCC, a multireference method, indicate that the transition states have little multireference character. Given the reasonable estimate of the barrier afforded by B97D-D3, and its tremendous performance advantage over SCS-MP2, CCSD(T) and MR-AQCC, this is the preferred method (at least with current technology) for examining Diels-Alder reactions like these, especially with larger molecules.

I recently blogged about a paper arguing that modern density functional development has strayed from the path of improving density description, in favor of improved energetics. The Medvedev paper1 was met with a number of criticisms. A potential “out” from the conclusions of the work was that perhaps molecular densities do not fare so poorly with more modern functionals, following the argument that better energies might reflect better densities in bonding regions.

The Hammes-Schiffer group have now examined 14 diatomic molecules with the goal of testing just this hypothesis.2 They subjected both homonuclear diatomics, like N2, Cl2, and Li2, and heteronuclear diatomics, like HF, LiF, and SC, to 90 different density functionals using the very large aug-cc-pCVQZ basis set. Using the CCSD density as a reference, they examined the differences in the densities predicted by the various functional both along the internuclear axis and perpendicular to it.

The 20 functionals that do the best job in mimicking the CCSD density are all hybrid GGA functionals, along with the sole double hybrid functional included in the study (B2PLYP). These functionals date from 1993 to 2012. The 20 functionals that do the poorest job include functionals from all rung-types, and date from 1980-2012. A very slight upward trend can be observed in the density error increasing with development year, while the error in the dissociation energy clearly is decreasing over time.

They note that six functionals of the Minnesota-type, those that are highly parameterized and of recent vintage, perform very poorly at predicting atomic densities, but do well with the diatomic densities.

Hammes-Schiffer concludes that their diatomic results support the general trend noted by Medvedev’s atomic results, that density description is lagging in more recently developed functionals. I’d add that this trend is not as dramatic for the diatomics as for atoms.

They pose what is really the key question: “Is the purpose to approximate the exact functional or simply to provide chemists with a useful tool for exploring chemical systems?” Since, as they note, the modern highly parameterized functionals have worked so well for predicting energies and geometries, “the observation that many modern functionals produce incorrect densities could be of no great consequence for many studies”. Nonetheless, “the ultimate goal is still to obtain both accurate densities and accurate energies”.

“Getting the right answer for the right reason” – how important is this principle when it comes to computational chemistry? Medvedev and co-workers argue that when it comes to DFT, trends in functional development have overlooked this maxim in favor of utility.1 Specifically, they note that

There exists an exact functional that yields the exact energy of a system from its exact density.

Over the past two decades a great deal of effort has gone into functional development, mostly in an empirical way done usually to improve energy prediction. This approach has a problem:

[It], however, overlooks the fact that the reproduction of exact energy is not a feature of the exact functional, unless the input electron density is exact as well.

So, these authors have studied functional performance with regards to obtaining proper electron densities. Using CCSD/aug-cc-pwCV5Z as the benchmark, they computed the electron density for a number of neutral and cationic atoms having 2, 4, or 10 electrons. Then, they computed the densities with 128 different functionals of all of the rungs of Jacob’s ladder. They find that accuracy was increasing as new functionals were developed from the 1970s to the early 2000s. Since then, however, newer functionals have tended towards poorer electron densities, even though energy prediction has continued to improve. Medvedev et al argue that the recent trend in DFT development has been towards functionals that are highly parameterized to fit energies with no consideration given to other aspects including the density or constraints of the exact functional.

In the same issue of Science, Hammes-Schiffer comments about this paper.2 She notes some technical issues, most importantly that the benchmark study is for atoms and that molecular densities might be a different issue. But more philosophically (and practically), she points out that for many chemical and biological systems, the energy and structure are of more interest than the density. Depending on where the errors in density occur, these errors may not be of particular relevance in understanding reactivity; i.e., if the errors are largely near the nuclei but the valence region is well described then reactions (transition states) might be treated reasonably well. She proposes that future development of functionals, likely still to be driven by empirical fitting, might include other data to fit to that may better reflect the density, such as dipole moments. This seems like a quite logical and rational step to take next.

A commentary by Korth3 summarizes a number of additional concerns regarding the Medvedev paper. The last concern is the one I find most striking:

Even if there really are (new) problems, it is as unclear as before how they can be overcome…With this in mind, it does not seem unreasonable to compromise on the quality of the atomic densities to improve the description of more relevant properties, such as the energetics of molecules.

Korth concludes with

In the meantime, while theoreticians should not rest until they have the right answer for the right reason, computational chemists and experimentalists will most likely continue to be happy with helpful answers for good reasons.

I do really think this is the correct take-away message: DFT does appear to provide good predictions of a variety of chemical and physical properties, and it will remain a widely utilized tool even if the density that underpins the theory is incorrect. Functional development must continue, and Medvedev et al. remind us of this need.

The optimized structures (B3LYP-D3BJ/def2-TZVPP) of these compounds are shown in Figure 1.

1

2

3

4

5

6

Figure 1. B3LYP-D3BJ/def2-TZVPP optimized geometries of 1-6.

Using the W1-F12 and W2-F12 composite methods, the estimated the heats of formation of these hydrocarbons are listed in Table 1. Experimental values are available only for 2 and 3; the computed values are off by about 2 kcal mol-1, which the authors argue is just outside the error bars of the computations. They suggest that the experiments might need to be revisited.

Table 1. Heats of formation (kcal mol-1) of 1-6.

compound

ΔHf (comp)

ΔHf (expt)

1

128.57

2

144.78

142.7 ± 1.2

3

20.18

22.4 ± 1

4

132.84

5

119.42

6

63.08

They conclude with a comparison of strain energies computed using isogyric, isodesmic, and homodesmotic reactions with a variety of computational methods. Somewhat disappointingly, most DFT methods have appreciable errors compared with the W1-F12 results, and the errors vary depend on the chemical reaction employed. However, the double hybrid method DSD-PBEP86-D3BJ consistently reproduces the W1-F12 results.

The role of dispersion in organic chemistry has been slowly recognized as being quite critical in a variety of systems. I have blogged on this subject many times, discussing new methods for properly treating dispersion within quantum computations along with a variety of molecular systems where dispersion plays a critical role. Schreiner1 has recently published a very nice review of molecular systems where dispersion is a key component towards understanding structure and/or properties.

In a similar vein, Wegner and coworkers have examined the Z to E transition of azobenzene systems (1a-g → 2a-g) using both experiment and computation.2 They excited the azobenzenes to the Z conformation and then monitored the rate for conversion to the E conformation. In addition they optimized the geometries of the two conformers and the transition state for their interconversion at both B3LYP/6-311G(d,p) and B3LYP-D3/6-311G(d,p). The optimized structure of the t-butyl-substituted system is shown in Figure 1.

The experiment finds that the largest activation barriers are for the adamantly 1f and t-butyl 1b azobenzenes, while the lowest barriers are for the parent 1a and methylated 1c azobenzenes.

The trends in these barriers are not reproduced at B3LYP but are reproduced at B3LYP-D3. This suggests that dispersion is playing a role. In the Z conformations, the two phenyl groups are close together, and if appropriately substituted with bulky substituents, contrary to what might be traditionally thought, the steric bulk does not destabilize the Z form but actually serves to increase the dispersion stabilization between these groups. This leads to a higher barrier for conversion from the Z conformer to the E conformer with increasing steric bulk.

Here is another fine example of the power of combining experiment and computation. Xu and co-worker has applied the FT microwave technique, which has been used in conjunction with computation by the Alonso group (especially) as described in these posts, to the trimer of 2-fluoroethanol.1 They computed a number of trimer structures at MP2/6-311++G(2d,p) in an attempt to match up the computed spectroscopic constants with the experimental constants. The two lowest energy structures are shown in Figure 1. The second lowest energy structure has nice symmetry, but it does not match up well with the experimental spectra. However, the lowest energy structure is in very good agreement with the experiments.

(0.0)

(4.15)

Table 1. MP2/6-311++G(2d,p) optimized structures and relative energies (kJ mol-1) of the two lowest energy structures of the trimer of 2-fluoroethanol. The added orange lines in the lowest energy structure denote the bifurcated hydrogen bonds identified by QTAIM.

Of particular note is that topological electron density analysis (also known as quantum theoretical atoms in a molecule, QTAIM) of the wavefunction of the lowest energy structure of the trimer identifies two hydrogen bond bifurcations. The authors suggest that these additional interactions are responsible, in part, for the stability of this lowest energy structure.

InChIs

The keto-enol tautomerization is a fundamental concept in organic chemistry, taught in the introductory college course. As such, it provides an excellent test reaction to benchmark the performance computational methods. Acevedo and colleagues have reported just such a benchmark study.1

First, the compare a wide variety of methods, ranging from semi-empirical, to DFT, and to composite procedures, with experimental gas-phase free energy of tautomerization. They use seven such examples, two of which are shown in Scheme 1. The best results from each computation category are AM1, with a mean absolute error (MAE) of 1.73 kcal mol-1, M06/6-31+G(d,p), with a MAE of 0.71 kcal mol-1, and G4, with a MAE of 0.95 kcal mol-1. All of the modern functionals do a fairly good job, with MAEs less than 1.3 kcal mol-1.

Scheme 1

As might be expected, the errors were appreciably larger for predicting the free energy of tautomerization, with a good spread of errors depending on the method for handling solvent (PCM, CPCM, SMD) and the choice of cavity radius. The best results were with the G4/PCM/UA0 procedure, though M06/6-31+G(d,p)/PCM and either UA0 or UFF performed quite well, at considerably less computational expense.

A 2013 study of oxalic acid 1 failed to uncover any tunneling between its conformations,1 despite observation of tunneling in other carboxylic acids (see this post). This was rationalized by computations which suggested rather high rearrangement barriers. Schreiner, Csaszar, and Allen have now re-examined oxalic acid using both experiments and computations and find what they call domino tunneling.2

First, they determined the structures of the three conformations of 1 along with the two transition states interconnecting them using the focal point method. These geometries and relative energies are shown in Figure 1. The barrier for the two rearrangement steps are smaller than previous computations suggest, which suggests that tunneling may be possible.

1tTt(0.0)

TS1(9.7)

1cTt(-1.4)

TS2(9.0)

1cTc(-4.0)

Figure 1. Geometries of the conformers of 1 and the TS for rearrangement and relative energies (kcal mol-1)

Placing oxalic acid in a neon matrix at 3 K and then exposing it to IR radiation populates the excited 1tTt conformation. This state then decays to both 1cTt and 1cTc, which can only happen through a tunneling process at this very cold temperature. Kinetic analysis indicates that there are two different rates for decay from both 1tTt and 1cTc, with the two rates associated with different types of sites within the matrix.

The intrinsic reaction paths for the two rearrangements: 1tTt → 1cTt and → 1cTc were obtained at MP2/aug-cc-pVTZ. Numerical integration and the WKB method yield similar half-lives: about 42 h for the first reaction and 23 h for the second reaction. These match up very well with the experimental half-lives from the fast matrix sites of 43 ± 4 h and 30 ± 20 h, respectively. Thus, the two steps take place sequentially via tunneling, like dominos falling over.